Algebra Examples

$(2, 3)$ perpendicular to $y = 1 - x$

Step 1

Find the slope when $y = 1 - x$ .

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.1.2

Reorder $1$ and $- x$ .

$y = - x + 1$

Step 1.2

Using the slope-intercept form, the slope is $- 1$ .

$m = - 1$

Step 2

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{- 1}$

Step 3

Simplify $- \frac{1}{- 1}$ to find the slope of the perpendicular line.

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Step 3.1

Divide $1$ by $- 1$ .

$m_{perpendicular} = 1$

Step 3.2

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1$

Step 4

Find the equation of the perpendicular line using the point-slope formula.

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Step 4.1

Use the slope $1$ and a given point $(2, 3)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = 1 \cdot (x - (2))$

Step 4.2

Simplify the equation and keep it in point-slope form.

$y - 3 = 1 \cdot (x - 2)$

Step 5

Solve for $y$ .

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Step 5.1

Multiply $x - 2$ by $1$ .

$y - 3 = x - 2$

Step 5.2

Move all terms not containing $y$ to the right side of the equation.

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Step 5.2.1

Add $3$ to both sides of the equation.

$y = x - 2 + 3$

Step 5.2.2

Add $- 2$ and $3$ .

$y = x + 1$

Step 6